An intrinsic approach to stable embedding of normal surface deformations

We introduce the notion of involutive Kodaira-Spencer deformations of the regular part X0 of a normal surface singularity, which form a subspace of the analytic cohomology H1(X0, T1,0X0). Examples of involutive deformations for which the Stein completion does not embed in a complex Euclidean space of stable dimension are in fact well-known. Under the assumption that X0 admits a Kähler metric with L2-curvature, we show that unstable deformations are avoided if the holomorphic functions which determine an embedding of the central fibre are correspondingly deformed into functions which can be uniformly bounded on compact subsets.